Make the outer circumference of the tire smaller
This is easy using proportional editing and the scale tool.

Go into an orthographic view that shows the tire from the front (it would roll towards the camera).
Set viewing mode to wireframe, with ghost mode on.
Use Border Select, Ctrl+B, to select the entire tread area of the tire.
Switch to the scale tool.
Scale along the desired plane (the two axes that are not the axle axes). The sides of the tire WILL NOT FOLLOW yet.
Go to the Adjust Last Operation area in the bottom right of the 3D View, and turn on Proportional Editing. Set the size to something just under the former distance between the tire treads and the part that contacts the wheel.
The walls of the tire should gradually tighten towards the treads, but stay mostly the same towards the wheel.